Material systems and methods of manufacture for auxetic foams

ABSTRACT

A novel material for producing auxetic foams is disclosed. The material comprises a multiphase, multicomponent polymer foam with a filler polymer having a carefully selected glass transition temperature. Novel methods for producing auxetic foams from the material are also disclosed that consistently, reliably and quickly produce auxetic polyurethane foam at about room temperature (25° C.). This technology overcomes challenging issues in the large-scale production of auxetic PU foams, such as unfavorable heat-transmission problem and harmful organic solvents.

REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of and claims priority toInternational Patent Application No. PCT/US2015/041713 filed Jul. 23,2015, entitled, “Material Systems and Methods of Manufacture for AuxeticFoams,” which claims priority to provisional U.S. Patent ApplicationSer. No. 62/029,225 filed on Jul. 25, 2014, entitled, “Auxetic Foams andMethods of Manufacture.” which is hereby incorporated by reference inits entirety.

BACKGROUND OF THE INVENTION Field of the Invention

This invention relates, generally, to polymeric foams. Morespecifically, it relates to polymeric foams exhibiting auxeticproperties and methods of manufacture.

BRIEF DESCRIPTION OF THE PRIOR ART

Auxetics refer to a family of materials possessing negative Poisson'sratio (ν) (the fraction of expansion divided by the fraction ofcompression for a material that is compressed in one direction whileexpanding in two other directions). [1-4] These materials expandlaterally during stretching, and shrink under compression. [1-4]Although such materials had been proposed in the literature at thebeginning of last century, they did not attract much attention at thattime, because they are rarely found in nature [5] In 1987, Lakes et al.first reported the manufacturing of artificial auxetic materials intheir pioneering work. [1] Their findings created significant interestin the development of auxetic materials because of the novel propertiesand promising application potential they exhibited, such as enhancedindentation resistance for applications in protective equipment [6-8],improved bending stiffness and shear resistance for structural integrityconstruction elements [9-13], optimal dynamics, acoustic and dielectricproperties for damping application and wave absorbers [14-17], etc.

A particular class of auxetic materials that has been developed is thatof auxetic polyurethane (PU) foams manufactured from conventionalflexible PU foams via a typical thermal mechanical process asillustrated in FIG. 1 . [1, 18-22] Essentially, this process, firstproposed by Lakes et al., involves applying triaxial compression on thepristine foam 105 to buckle the cell struts and induce the re-entrantmorphology. The compressed foam was then heated above the softeningtemperature of the polymer, followed by cooling in the compression stateto fix the intended re-entrant structure 110. [1,23,24]

In the past, there have been many efforts to explore the effects offabrication parameters on the structure and properties of auxetic PUfoams, in order to control the manufacture process more efficiently.[1,10,20,24-30] Also, attempts have been made to develop a scale-upmanufacture process. [24,31] One might conclude that the fabricationprocess is no longer the obstructive element in the development ofauxetic PU foams in regard to the apparent simplicity of the overprinciple in the manufacture process, and great effort devoted in thisfield. However, this statement should be considered with caution,because large discrepancies are found in the reports given by variousauthors. For example, published studies have shown that both processingtemperature and heating time varied in extremely wide ranges of 130-220°C. and 6-60 min respectively, while no connections between them wereobserved (see a recent review by Critchley et al. 1321). It has beenargued that the discrepancies can be attributed to the variances inequipment employed in each individual research and macroscopic cellularstructure (e.g., cell size) of PU foams. Surprisingly, few studies haveconsidered the variance in chemistry and microstructure of originalmaterials as a critical factor.

Still, although the manufacturing of auxetic PU foams has been known foralmost three decades, the fabrication of auxetic PU foams is atime-consuming, trial-and-error, and user-dependent process. Severalcritical issues remain unresolved. The criteria for the selection ofsuitable flexible PU foams for auxetic manufacturing have not beenestablished. The methodologic principle relevant to defining thesoftening temperature is not yet clear. The identification of optimumheating temperature and heating time is still the subject of controversybetween authors.

The manufacture of auxetic PU foams from conventional PU foams generallyinvolves three steps: volumetric compression, followed by heating beyondthe softening temperature and then cooling in the compression state.[10,18-20,22,24,26-30,32] Although this approach is simple and theprocedure is convenient to operate, it suffers from an inherentheat-transmission problem due to thermal insulation properties PU foamsexhibit (thermal diffusivity, α, ˜1×10⁻⁷ m²/s-9×10⁻⁷ m²/s) [43]. Thisproblem can be illustrated by considering the characteristic heatingtime t=l²/α, which shows significant increase when sample size (1)increases. [57] Another drawback of this method is that the auxeticfoams produced using this approach show a non-uniform microstructure andirregular properties because of the existence of unavoidable temperaturegradient during manufacture. Thus this thermal-mechanical techniqueappears to be difficult to scale-up for commercial applications. On theother hand, the chemical-mechanical method relying on acetone has beenproposed by Grima et al. [55] Auxetic PU foams were fabricated byplacing compressed PU foams in acetone at room temperature and then airdrying. It was discovered that such method can produce more homogeneousauxetic foams and enables creation of larger samples in technique.[55,56] However, this technique is of limited practical use because thistechnique requires large amounts of volatile organic solvent asprocessing aids, which needs to be subsequently removed. The completeremoval of the solvent, which is essential for the product performance,is both time and energy intensive. All these add a great deal of costand process complexity.

SUMMARY OF THE INVENTION

Various embodiments may comprise a method for producing an auxetic foam.A flexible foam may be provided, which may have an initial volume andmay comprise a plurality of cells. The flexible foam may comprise a softdomain, a hard domain, and a filler polymer. The foam may be placed intoa pressure chamber. The foam may be compressed to a compressed volumethat is less than the initial volume. The cells of the foam may bedeformed when the foam is compressed. The compressed foam may be exposedto a compressed gas within the pressure chamber. The pressure chambermay be maintained at a predetermined temperature and pressure for apredetermined time. At least a portion of the compressed gas maydissolve into the filler polymer. The dissolved gas may reduce a glasstransition temperature of the filler polymer such that the fillerpolymer transitions from a glassy state to a rubbery state. Thereafterthe pressure may be relieved and allowed to reach atmospheric pressurebefore removing the foam from the chamber, such that the filler polymertransitions from the rubbery state to the glassy state, thereby fixingthe cells of the foam in the deformed state.

Additional embodiments may comprise a method for producing an auxeticfoam. A flexible foam may be provided. The flexible foam may have aninitial volume and may comprise a plurality of cells. The flexible foammay comprise a soft domain, a hard domain, and a filler polymer. Thefoam may be placed into a pressure chamber. The foam may be compressedto a compressed volume that is less than the initial volume. The cellsof the foam may be deformed when the foam is compressed. The compressedfoam may be exposed to carbon dioxide within the pressure chamber. Thepressure chamber may be maintained at a predetermined temperature andpressure for a predetermined time. At least a portion of the carbondioxide may dissolve into the filler polymer. The dissolved carbondioxide may reduce the glass transition temperature of the fillerpolymer such that a shape of the filler polymer transitions fromgenerally spherical to generally ellipsoidal. Thereafter the pressuremay be relieved and allowed to reach atmospheric pressure beforeremoving the foam from the chamber, such that the filler polymer retainsthe generally ellipsoidal shape, thereby fixing the cells of the foam inthe deformed state.

Still further embodiments may comprise a method for producing an auxeticfoam. A flexible foam may be provided. The foam may have an initialvolume and may comprise a plurality of cells. The flexible foam maycomprise a soft domain having a first glass transition temperature, ahard domain having a thermal transition temperature greater than thefirst glass transition temperature, and a filler polymer. The filler mayhave a second glass transition temperature greater than the first glasstransition temperature and less than the thermal transition temperature.The foam may be placed into a pressure chamber and compressed to acompressed volume that is less than the initial volume. The cells of thefoam may be deformed when the foam is compressed. The compressed foammay be heated within the pressure chamber to a temperature greater thanthe second glass transition temperature and less than the thermaltransition temperature. The compressed foam may be held at thetemperature for a predetermined time. The compressed foam may be allowedto cool while remaining in the pressure chamber to a temperature lessthan the second glass transition temperature. The foam may be removedfrom the pressure chamber, thereby fixing the cells of the foam in thedeformed state.

Yet other embodiments may comprise a material system for the productionof auxetic foams. The material system may comprise a bulk matrix whichin turn comprises a soft domain comprising a polymer chain and having afirst glass transition temperature. The bulk matrix may further comprisea hard domain covalently or non-covalently linked to the polymer chainof the soft domain and having a thermal transition temperature greaterthan the first glass transition temperature. The material system mayalso comprise a filler polymer dispersed in the bulk matrix. The fillerpolymer may have a second glass transition temperature greater than thefirst glass transition temperature and less than the thermal transitiontemperature.

BRIEF DESCRIPTION OF THE DRAWINGS

For a fuller understanding of the invention, reference should be made tothe following detailed description, taken in connection with theaccompanying drawings, in which:

FIG. 1 is a schematic illustration of the structure conversion fromconventional foams to auxetic foams.

FIG. 2 illustrates a schematic diagram of auxetic foam manufacture.

FIG. 3 is a schematic diagram of a video extensometer measurement ofPoisson's ratio.

FIG. 4 is a diagram of the microstructures and compositions of flexiblepolyurethane foams.

FIG. 5 illustrates various cross-sectional SEM images of flexible PUfoams.

FIG. 6A is a graph of the ATR-F FIR spectra of flexible PU foams.

FIG. 6B is a graph of the FTIR spectra of the extractions of flexible PUfoams by DMF.

FIG. 7 is a graph of DSC thermograms for flexible PU foams.

FIG. 8A is a graph of the loss factor (tan δ) peak.

FIG. 8B is a graph of the normalized dynamic storage modulus for PUfoams at 1 Hz and 1° C./min from −100° C. to 200° C.

FIG. 9A is a graph of SAXS scattered intensity profiles.

FIG. 98 illustrates WAXS patterns.

FIG. 9C is a graph of WAXS scattered intensity profiles for threeflexible PU foams at 25° C.

FIG. 10 is a stress-strain curve of polyurethane foams in compression ata strain rate 0.01 min⁻¹.

FIG. 11 is a graph of structural convertibility of the three foams at135° C.

FIG. 12 is a SEM image of foam II after 70% compression at 135° C. for30 min.

FIG. 13 is a schematic representation of basic deformation mechanism inSAN-containing polyurethane foams.

FIG. 14A is a graph of structural convertibility curve of PU foam II atdifferent shape holding temperatures (shape strain 40%).

FIG. 14B is a graph of structural convertibility of PU foam III atdifferent shape holding temperatures (shape strain 40%).

FIG. 14C is a plot of log[−ln(1-R_(f))] vs. log(t) of PU foam for foamII. The dash lines are the KWW stretched exponential hits.

FIG. 14D is a plot of log[−ln(1-R_(f))] vs. log(t) of PU foam for foamIII. The dash lines are the KWW stretched exponential fits.

FIG. 14E is a plot of relaxation time as a function of temperature forPU foam II.

FIG. 14F is a plot of relaxation time as a function of temperature forPU foam III.

FIG. 15 is a graph showing heating time as a function of processtemperature in the auxetic manufacture of foams II and III. Insetnumbers are measured mean Poisson's ratio of auxetic foams.

FIG. 16A is a SEM image of the auxetic foam prepared from foam H at 150°C. for 20 min (volume compression ratio 0.7).

FIG. 16B is a SEM image of the auxetic foam prepared from foam III at150° C. for 20 min (volume compression ratio 0.7).

FIG. 16C illustrates that SAN particles are stretched along the localstress direction.

FIG. 16D illustrates that SAN particles are stretched along the localstress direction.

FIG. 17A presents a SEM image of the sample prepared from foam III at150° C. for 60 min (volume compression ratio 0.85).

FIG. 17B presents a SEM image of the sample prepared from foam III at150° C. for 60 min (volume compression ratio 0.85).

FIG. 18 is a schematic diagram of the process window for the fabricationof auxetic flexible PU foam.

FIG. 19 is a schematic diagram of sample preparation equipment for thefabrication of auxetic PU foams with compressed CO₂.

FIG. 20A illustrates a temperature and pressure processing window forauxetic manufacture of PU foams with compressed CO₂. The circle anddiamond show unsuccessful and successful processing conditions,respectively (the inset values are measured Poisson's ratio). The linein the graph is the glass transition temperature (T₅) of SAN (ANcontent, 30 wt %)—CO₂ system.

FIG. 20B is a graph of Poisson's ratio values as a function of axialengineering strain for PU foam and auxetic PU foam fabricated at 25° C.and 5 MPa with a predefined volumetric change 0.75. The inset in FIG.20B shows the missing-rib unite cell.

FIG. 20C presents scanning electron micrographs demonstrating themorphology of raw PU foam and auxetic PU foam fabricated at 25° C. and 5MPa with a predefined volumetric change 0.75.

FIG. 20D is a schematic of a mechanism for structural change in PU foamsduring auxetic conversion.

FIG. 21A is a graph of the effect of processing time of auxetic PU foamsduring auxetic manufacture with compressed CO₂. Auxetic samples werefabricated at room temperature (25° C.) and very modest pressure (5 MPa)with a predefined volumetric change 0.75. The inset in FIG. 21A shows aschematic diagram of CO₂ transport in a representative elementary volumeof PU foam.

FIG. 21B is a graph of the effect of volumetric change on Poisson'sratio of auxetic PU foams during auxetic manufacture with compressedCO₂. Auxetic samples were fabricated at room temperature (25° C.) andvery modest pressure (5 MPa) with a predefined volumetric change 0.75.

FIG. 22 is a schematic diagram of a method for producing auxetic foam.

FIG. 23A illustrates the shape of auxetic foam in sheet form when bent.

FIG. 23B illustrates the shape of non-auxetic foam in sheet form whenbent.

FIG. 24 is a schematic diagram of a method for producing a prostheticsock comprised of auxetic foam.

FIG. 25 is a schematic diagram of a method for producing a prostheticsock comprised of auxetic foam.

FIG. 26 is a flow chart of an exemplary method for producing an emeticfoam.

FIG. 27 is a flow chart of an exemplary method for producing an auxeticfoam.

FIG. 28 is a flow chart of an exemplary method for producing an auxeticfoam.

FIG. 29 is a graph of glass transition behavior as a function ofpressure for pure SAN.

FIG. 30 is a graph showing a comparison of experimental solubility andpredicted solubility.

FIG. 31 is a graph showing the effect of volumetric change on Poisson'sratio of auxetic PU foams during auxetic manufacture with compressedCO₂.

FIG. 32 is a graph showing stress-strain plots from the compression testof foams with different volumetric change ratios.

FIG. 33 is a graph showing energy absorption (W) versus peak stresscurve of foams with different volumetric change ratios.

FIG. 34 is a graph showing cushioning coefficient (C) versus stresscurve of foams with different volumetric change ratios.

FIG. 35 is a graph showing energy dissipation of foams with differentvolumetric change ratios (and Poisson's ratios).

FIG. 36 is a graph showing dynamic properties of raw PU foam and auxeticPU foam.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Various embodiments as disclosed herein may comprise material systems asthe starting materials for auxetic foam fabrication. The material systemmay comprise multiphase, multicomponent open cell polymer foams. Furtherembodiments may comprise methods for producing auxetic foams using thesestarting materials.

A foam may comprise two parts: a bulk matrix polymer part (non-porouspart) and a voided part (the porous part or the space (cells) containedin the foam). The bulk matrix of the system may comprise a hard domainproviding strength and a soft domain providing deformation capability.For a PU foam (produced by reacting an isocyanate with a polyol), thesoft domain may comprise long polymer segments derived from the polyols.The longer the soft domain polymer, the lower the force required todeform the PU foam, and the larger deformation the PU foam can undergo.The hard domain may comprise isocyanate covalently or non-covalentlylinked to the soft domain polymer. Shorter soft domain polymers withhigher levels of crosslinking leads to more rigid and tougher foams.

The bulk matrix may further comprise a third component, referred to as afiller polymer, as a dispersed phase (third domain) in the bulk matrix.The filler polymer may have different mechanical properties and may havedifferent glass transition temperatures that all lie between those ofthe hard and soft domains. In various embodiments, the type andconcentration of the filler polymer in the bulk matrix may be varied totailor mechanical and thermal properties of the final auxetic foam. Thedeformation of the filler polymer under temperature-time history maydictate the fixation of the auxetic foam structure and the Poisson'sratio of the foam.

Depending on the combination of the three primary components (i.e., harddomain, soft domain, and filler polymer), auxetic foams with a widerange of mechanical properties (such as strength, modulus, andelongation) and Poisson's ratio may be produced. In addition, thethermal stability of the auxetic foam may be varied by the primarycomponent selection.

In various embodiments, the filler polymer may comprise, for example,styrene acrylonitrile (SAN), polyether sulfone (PES), polysulfone,cyclic olefin copolymers (COC), acrylonitrile-butadiene-styrene (ABS),polyp-phenylene oxide (PPO), poly(ether ketone (PEK), poly(ether etherketone) (PEEK), poly(ether ketone ketone) (PEKK) or mixtures thereof.

Auxetic foams may be produced by first mechanically compressing thematerial system (PU foam) to a predetermined volume compression ratio.The mechanical compression deforms the cells of the foam. Whilemaintaining compression, the material system may heated to a temperatureabove the glass transition temperature of the filler polymer and below athermal transition temperature (which may be, for example, a glasstransition temperature, a melting temperature, or a solidificationtemperature) of the hard domain, allowing the filler polymer totransition from a glassy state to a rubbery state which in turn allowsthe filler polymer to deform. The material system may remain at thedesired temperature for a period of time, then allowed to cool to roomtemperature. As the material system cools below the glass transitiontemperature of the filler polymer, the filler polymer transitions backto the glassy state which permanently fix the shape of the fillerpolymer. This in turn fixes the cell deformation and overall foam shape.The mechanical compression is then released. The material produced is anauxetic foam resulting from the filler polymer fixing the foam cells inthe deformed state when the filler polymer transitions back to theglassy state. The deformation of the filler polymer undertemperature-time history may dictate the fixation of the auxetic foamstructure and the Poisson's ratio of the foam.

Additional embodiments may comprise subjecting the material system to acompressed gas that can be dissolved from a few tenths of a percent byweight to several tens of percent by weight, resulting from favorablepolymer-gas intermolecular interaction. Exemplary gases include carbondioxide, nitrogen, or any volatile organic chemical. The dissolved gasgenerates additional free volume and increase the mobility of thepolymer chains of the filler polymer, resulting in a reduction of theglass transition temperature of the filler polymer, whose value can beeither experimentally measured or calculated theoretically. In additionto the list of exemplary filler polymers presented above, the fillerpolymer may comprise any polymer in which the compressed gas hassubstantial solubility.

Because the glass transition temperature of the filler polymer is theminimum temperature required to allow the filler polymer to be deformedto new permanent shapes under mechanical compression, and is depressedby the dissolved gas, the material system may be heated to a lowertemperature than would be necessary without dissolving the gas in thefiller polymer, to cause the filler polymer to deform under mechanicalcompression and relax to the deformed shape. Moreover, because of thegreatly increased polymer chain mobility resulting from the dissolvedgas, polymer chain relaxation under mechanical stress, which is themolecular mechanism for the filler polymer to achieve new shapes undermechanical compression, can be greatly accelerated. That is to say underthe same temperature, the time needed to reach the same deformation offiller polymer is much shorter with dissolved gas than would needwithout the dissolved gas. Therefore by using compressed gas it ispossible to manufacture auxetic foams at lower temperature with reducedamount of time. Both are of tremendous benefit for large scalemanufacturing of emetic foam in a cost effective way. Polymers andpolymer foams are insulators, and heating them up takes a long time andconsumes a large amount of energy, most of which are wasted during thelengthy heating process. Reduced requirement for the manufacturingtemperature therefore leads to significant energy savings andmanufacturing time reduction. Moreover, because the polymer foams andpolymers are very poor in heat transfer, a higher manufacturingtemperature not only consumes more energy, but also leads to more nonuniform temperature distribution throughout the foams. Thenon-uniformity becomes more severe when the product size becomes larger.Such temperature non-uniformity has extreme detrimental effect on theproduct quality, ultimately limiting the size of the product that can beproduced (typically very small and very thin). By lowering themanufacturing temperature by using compressed gas, this difficulty canbe greatly alleviated, which allows for manufacturing of larger sizeproduct. The type of the compressed gas and the pressure can bejudicious selected for the particular filler polymer, such that themanufacturing temperature may be close or equal to room temperature,completely eliminating the need for heating and associated equipmentcost. Furthermore, since the compressed gas may have high diffusivityand very short distance to diffuse from the outer surface of the matrixpolymer to the filler polymer within, the manufacturing time can befurther reduced to minutes or even seconds. For example, for a PU foamsystem in which SAN is the filler polymer, we have used carbon dioxideas the compressed gas, and were able to manufacture auxetic foam withPoisson's ratio of −0.5 at room temperature using a manufacturing timeas short as less than 1 minute. See FIG. 21A, and the discussion belowfor details.

Chemistry and Structure of Auxetic Foams

Three commercial open-cell, flexible PU foams with nominal cell diameterof 480 μm were obtained (referred to herein as I, II and III). Thedensities of the foams were 44.8 kg/m³, 44.8 kg/m³ and 48.1 kg/m³,respectively. They were dried in an air-flow oven at 80° C. for at least12 h before use. Auxetic foams were fabricated by a thereto-mechanicalprocess, as illustrated in FIG. 2 . [19,22] Samples of the foam 205 withinitial dimensions of 32 mm diameter (d₀) and 80 mm length (h₀) wereplaced into a metallic tube 210 (i.e., a pressure chamber) of mold 200and compressed by a piston 215 to 20 mm diameter (d₁) and 50 mm length(h₁) as shown in FIG. 2 . The mold 200 was then placed into an oven withforced convection, of which the temperature can be controlled with anaccuracy of 0.3° C. (FED 115, Binder GmnbH, Germany) at test conditionsfor a constant time. The mold 200 was then removed from the oven andcooled at room temperature for 1 h. Finally, the foams 205 were takenout of the mold and stretched in the axial directions. The stiffer endsof the specimens 205 were cut.

Sol/Gel Analysis

Foam samples 205 approximately 2 g were immersed in 500 ml ofdimethylfomamide (DMF). After 48 h, the solvent swollen samples 205 wereremoved from the DMF/sol fraction solutions and dried in a vacuum ovenat 40° C. for 24 h and then at 80° C. for an additional 24 h. The driedextracted samples 205 were then weighed to determine the sol fraction ofeach sample 205. Values of the soluble fraction for foams I, II and IIIfrom the solvent extraction experiments were found to be 5 wt %, 9 wt %and 18 wt % respectively.

Scanning Electron Microscope (SEM)

The morphologies of foam samples 205 were investigated using fieldemission scanning electron microscope (SEM) (JEOL 7401F). Samples 205were cut using a knife and the fracture surface was sputter-coated witha thin layer of gold before observation.

Infrared Analysis

Fourier transform infrared (FTIR) spectra were performed by a NicoletNEXUS 470 FTIR-spectrometer (Thermo Ltd.) with the KBr pellet techniquein a range from 4000 to 400 cm⁻¹ at a resolution of 4 cm⁻¹. Data werecollected as average of 32 scans. FTIR with attenuated total reflectance(ATR) spectra, were carried out in a spectral range from 4000 to 650cm⁻¹ utilizing a Smart Golden Gate reflectance attachment and recorded64 scans at a resolution of 2 cm¹. All spectra had been normalized usingthe CH₂ peak at 1969 cm⁻¹ as an internal reference peak.

Dynamic Mechanical Analysis (DMA)

The 7 mm×7 mm×25 mm rectangular foam samples 205 were machined using aCO₂ laser (VersaLASER, Universal Laser Systems). DMA was studied by a TAInstruments Q800 Dynamic Mechanical Analyzer in tension model using adeformation of 0.2% strain, a frequency of 1 Hz, a force track of 150%,and a preload force of 0.05 N. The test was run in the temperature rangeof −100 to 200° C. using a heating rate of 1° C./min.

X-ray Scattering

Simultaneous small- and wide-angle X-ray scattering (SAXS/WAXS)measurements were obtained using a Bruker NanoSTAR system, operating at45 kV and 650 mA with 1 μs microfocus X-ray source (the wavelength of CuKα, λ=0.15412 nm). The SAXS pattern was recorded by a HiStar 2Dmulti-wire area detector. The WAXS pattern was recorded by a Fuji PhotoFilm image plate and the plate was read with a Fuji FLA-7000 scanner. Inthe WAXS measurement, the distance between the image plate and thesample stage was 50 mm. A 600s exposure time was used for collecting theSAXS and WAXS scattering patterns for samples. The foam samples 205 werecut 6 mm thickness and compressed to 2 mm thickness.

Uniaxial Compression Testing

The compression experiments were conducted using a TA Instruments Q800Dynamic Mechanical Analyzer in compression mode with a 15 mm compressionclamp at a strain rate of 0.01 mini and 30° C. The disk foam sample 205which was 15 mm in diameter and 5 mm thick was machined using a CO₂laser (VersaLASER, Universal Laser Systems).

Structural Convertibility Characterization

Structural convertibility properties of the foam 205 were quantified viastrain-controlled compression tests performed on a ARES-LS3 rheometerwith 25 mm parallel plate fixture (TA instruments). The disk foamsamples 205, with a diameter of 25 mm and a thickness of 10 mm, weremachined using a CO₂ laser (VersaLASER, Universal Laser Systems). Thesamples 205 were then heated to the testing temperature and allowed toequilibrate for 10 min, and then compressed to a strain of 40% or 70% ata rate of approximately 0.5 mine. The compressed samples 205 were thenallowed to equilibrate at the testing temperature for different time.Auer cooling with strain at room temperature for an additional 10 min,the samples 205 were removed from the fixture and stored for 24 h toallow for the completion of the fast relaxation process. Finally, thesample thickness was measured, and the structural convertibility (Rd wascalculated using Equation 1:

R _(t)=ε/ε_(load)  Eqn. 1

where, ε is the strain after unloading and ε_(load) is the initialloading strain. Values averaged from three separate measurements wereused for calculation.

Measurement of Poisson's Ratio

Measurement of Poisson's ratio of the foam samples 205 was based onvideo data acquired from a video extensometer system (Shimadzu DV-201)machine 300 in a tensile test. The foam sample 205 was coupled to a topclamp 305 and a bottom clamp 310 of the video extensometer system 300.Two spaced apart lines 315 were placed on the foam sample 205perpendicular to a direction of travel of the clamps 305, 310. Thetensile test was conducted using a strain rate of 6 mm/min and maximumstrain of 10% (see FIG. 3 ). For the calculation of Poisson's ratio,video data obtained by camera 320 was first transformed into image datavia the software MATLAB R2013b. Secondly, a MATLAB routine was used tocalculate the length (l) and diameter (d) of the sample for every image.Thirdly, the transverse strain (ε_(x)) and longitudinal strain (ε_(y))were calculated, respectively, using Equations 2 and 3:

$\begin{matrix}{\varepsilon_{x} = \frac{\Delta l}{l_{0}}} & {{Eqn}.2}\end{matrix}$ $\begin{matrix}{\varepsilon_{y} = \frac{\Delta d}{d_{0}}} & {{Eqn},3}\end{matrix}$

where, l₀ is the original length and d₀ is the original diameter.Finally, the average Poisson's ratio was calculated from thestrain-strain curve by the classical definition of Poisson's ratio 1341given in Equation 4:

$\begin{matrix}{v = \frac{\varepsilon_{x}}{\varepsilon_{y}}} & {{Eqn}.4}\end{matrix}$

Effects of Chemistry and Structure on Auxetic Foam Production

Flexible PU foams 205 are chemically and structurally complex polymersformed from two competing reaction between a diisocyanate and both polyof and water. [35,36] FIG. 4 illustrates a typical two-level structureof flexible PU foams 205, which has been supported in numerous studies.[37-41] Besides the obvious macroscale cellular structure, flexible PUfoams 205 show a dominant phase-separated microscopic structure 405,consisting of hard domains dispersed in a soft domain covalent matrix.Two additional structures can be sometimes observed in flexible PU foams205. One is the large urea-rich aggregates or “urea balls” structurefrequently found in flexible PU foams 205 prepared with high waterconcentration. [35,36] The other is the styrene and acrylonitrilecopolymer (SAN) particles filled for the improvement of the load bearingproperty and cell openness of flexible PU foam 205. [41]

The structure of the flexible PU foams 205 was studied using SEManalysis. FIG. 5 shows the representative cross-sectional SEMmicrographs of PU foams 205. All foams 205 presented similar cellularstructure. However, a clear difference was found in SEM images taken athigh magnification. The rib surface of both samples II and III appearedwell dispersed spherical particles (˜0.5 μm) While the rib surface of Iwas smooth. This difference can also be observed in samples treated byDMF. It is clear that in place of microspheres presenting in originalfoams 205 there are holes with a size similar to particles in bothsamples II and III, while no difference was found in sample I.

FIG. 6A shows the ATR-FTIR spectra of the original foams 205. In the NHstretching vibration region (from 3,150 to 3,500 cm⁻¹), all foams 205show a peak around 3,295 cm⁻¹, which indicates the NH group forms ahard-soft domains hydrogen bond with ether oxygen (NH—O). [37] Thisindicates the existence of phase mixing between hard and soft domains.In the carbonyl (C═O) stretch region (from 1,600 to 1,800 cm⁻¹), allsamples show two peaks. One is the peak at 1.720 cm⁻¹ which is assignedto the free urethane C═O. The other is the peak around 1.642 cm⁻¹ whichindicates that most of urea C═O groups are associated in the orderedhard-hard domains hydrogen bond (NH—O═C). [37-39] This seems to contrastthe analysis from the NH stretching; however, this discrepancy can, atleast in part, be explained by coexistence of a homogeneous hard-softdomain mixing phase and ordered hard domains. Also notable is the muchhigher relative intensity of the ordered NH—O═C peak in III. Thisimplies that III has a higher content of hard domains. An importantpoint to notice in the spectra, is the nitrile (CN) peak ofacrylonitrile centered at 2,240 cm⁻¹, which is clearly visible in II andIII but not in I. This is a direct evidence for the presence of SANfillers in II and III (SAN content: III>II).

Furthermore, as presented in FIG. 68 , the FTIR analysis of extractionsby DMF shows that the amount of extractable SAN increases with theincreasing of SAN loading in flexible PU foams 205. It can also beobserved that the amount of the soluble hydrogen-bonded C═O (hardsegment) in foams 205 follows the sequence III>II>I. These resultssuggest that the links of these SAN copolymers and hard domains to thesoft domain matrix, which depend not only on chemical grafting, but alsoon physical means, such as hydrogen bonding. [40,41]

FIG. 7 shows the DSC curves of these foams 205. Results show that nocrystallinity was observed up to 200° C. This reveals that there are nosigns of the presence of large size urea aggregates in these foams 205.All foams 205 show a soft domain glass transition temperature of about−50° C. (denoted as T_(g·soft)). [41] For II and III, a SAN glasstransition temperature of about 105° C. (denoted as T_(g·SAN)) wasobserved. This agrees well with previous reports. [40-42]

To further probe the thermal mechanical properties of flexible PU foams205, the DMA technique was applied. The loss factors (tan δ) wereplotted as a function of temperature at a frequency of 1 Hz fortemperature ranging from −100° C. to 150° C. (FIG. 8A). The plotrelative to sample 1 exhibits a strong sharp peak at −38.7° C. (denotedas T_(g·soft)) which is attributed to the glass transition of softdomains. It is noted that this glass transition is greater than a puresoft polyol system (−55° C. by DMA at 1 Hz) [41], meaning a certaindegree of hard- and soft-domain mixing which significantly restrict thechain mobility of soft domains. For II and III, two distinct peaks areobserved in the tan δ curves: one is similar to that of I, which isascribed to the soft domain glass transition (denoted as T_(g·soft)),and the other is at much higher temperature which is attributed to theSAN reiteration process (denoted as T_(g·SAN)). This is in reasonableagreement with the DSC results, although they are about 10-15° C. higherthan the calorimetric values.

Deconvolution of the effects of the cellular structure on the elasticityof materials was done by varying the storage modulus, but maintainingthe value of the starting point in storage modulus curve at a constantvalue of 3×10⁹ Pa. [40,43,44] The normalized storage modulus plots (FIG.8B) show that the presence of SAN strongly increases the rubber plateaustorage modulus. This illustrates the classical reinforcement effect offiller polymers. Thus, the volume fraction (c) of SAN for each sample inthis study was estimated on the basis of the theory of filler polymerreinforcement to be 0 vol % (foam I), 11.8 vol % (foam II) and 31.6 vol% (foam III), using the Guth equation [45] (Equation 5):

E*=E ₀(1+2.5c+14.1c ²)  Eqn. 5

where E₀ is defined as E(T_(g·SAN)+20 K) and E* is defined asE(T_(g·SAN)−20 K).

To complete the microstructure picture in flexible PU foams 205,simultaneous small and wide-angle scattering measurements were alsoemployed. FIG. 9A shows the SAXS profiles for three foams 205 varying inthe SAN particle content. All samples 205 display only a shoulder at0.5-1 nm⁻¹ in the SAXS profiles. These results clearly suggest thatthere is the existence of phase separation, and also they provideindirect evidence for the weak interconnectivities between hard domains.[46] FIG. 9B shows the 2D WAXS patterns for the three foams 205. Only anamorphous halo at 2θ˜20° was observed, which suggest that very littleorder exist in the hard domains of foams 205. [47] The hard domain sizeestimated by the correlation length l) using Scherrer equation (l)≈kλ/βcos θ), [49] was found to be about 1 nm. This is far less than thesmallest structural heterogeneity resolvable through T_(g) measurements(˜10 nm), [49] where k is a material parameter, commonly 1 for polymer,λ is the wavelength of the X-ray, and β is the full width athalf-maximum of the WAXS peak (FIG. 9C). Note that from both SAXS andWAXS results, no noticeable difference in the three foams 205 can befound. This reveals that the incorporation of large size SAN particleson the phase separation structure is limited. Thus, the X-ray scatteringexperiments provided strong evidence of microphase separation structurein flexible PU foams 205 and a satisfying interpretation of the changesof soft domain T_(g).

The analysis has so far shown a relative complete understanding on thechemistry and structure of these foams 205 by the combination ofdifferent techniques. It can be concluded that all flexible PU foams 205exhibit complex phase separation structure with a large fraction of softdomains dissolved within the hard domains. It also shows that nolarge-size urea-rich aggregates are detected in these foams 205. For theSAN-containing foams (11 and III), SAN filler polymers are embedded in asoft domain matrix by both chemical and physical crosslinkings.

Structural Convertibility

FIG. 10 presents typical compressive stress-strain curves for theflexible foams 205. It is found that the overall trend of thestress-strain curve is similar for all foams 205. Beyond an initialelastic region (<10% strain), they show a long collapse plateau regionassociated with buckling of the cell struts. This is followed by adensification regime at 60% strain in which the cell struts begin tocontact with each other, resulting in a large upturn in stress. [43]FIG. 10 also illustrates the effect of SAN on the compressionproperties. In addition to the higher firmness, the SAN containing foams205 also exhibit greater hystersis, suggesting greater load relaxation.[50]

Two typical preloaded compressive strains of 40% (in collapse plateauregion) and 70% (in densification regime) were selected for the study ofstructural convertibility. As shown in FIG. 11 , the very distinctstructural convertibility (R_(f)) was observed, with the increase of theSAN content and preloaded strain. In the case of I with low preloadedstrain level (40%), the structural convertibility is directly ascribedto the hydrogen bonds. These bonds weaken as temperature increases andhard domain phase moves relative to the soft domain phase. When thetemperature is reduced, the hard and soft domain phases will resumetheir hydrogen bonding interactions in the new deformed geometry.However, these forces are normally weak relative to the overall strengthof the elastic matrix's restoring force, and also are easily affected byhumidity. [39]

In addition to the hydrogen bonds, an important factor introduced forthe structural conversion of II and III, is related to the temperaturedependence of the SAN chain mobility or the relaxation process of SANchains. These are quite evident from the SEM observation (FIG. 12 ). Itis proposed that above the glass transition temperature, the SANreinforcing filler polymers 1305 can relax quickly to be deformed fromtheir general spherical shapes (as shown in FIG. 5 ), into generallyellipsoidal shapes (as shown in FIG. 13 ), in response to external forceequilibrium. Cooling down below the glass transition temperature of SANparticles leads to vitrification of the new shape. Consequently,enhanced structural conversion would be expected. This is shownschematically in FIG. 13 .

Also notable is higher structural convertibility values at higherpreloaded strain for flexible PU foams 205. This enhanced structuralconvertibility might be assigned to the additional weak van der Waalsinteractions (“adhesion”) between the cell surfaces, which is directlyproportional to the surface contact areas between the cell ribs. Inother words, this interaction can be considered to be negligible beyondthe densification region as mentioned above.

Following the discussion above, it is perhaps surprising that thestructural convertibility of flexible PU foams 205 is associated withdifferent mechanisms which vary with the structure of foams and theprocessing conditions, e.g., temperature and strain dependence. Howevereven more surprising is the result that the SAN particles filled inflexible PU foams 205 play the most important role in the structuralconversion of these foams 205.

It is thus essential to further elucidate the effect of SAN relaxationprocess on the structural convertibility of SAN containing foams 205.For this purpose, systematic experimental measurements were performedwith II and III at different temperatures. The preloading strain usedfor each of the foams 205 was taken as 40% in which the adhesion effecton the structural convertibility can be neglected. As shown in FIGS. 14Aand 14B, a longer heating time gives a higher structural convertibilityand an increased temperature also favors a higher structuralconvertibility. The accompanying FIGS. 14C and 14D are replotting thesame results in FIGS. 14A and 14B as log[−ln(1-R_(f))] vs. log(f) indouble-logarithmic plots respectively, based on the empiricalKohlrausch, Williams and Watts (KWW) stretched exponential function.[53,54]. Equation 6 presents a function used to describe the structuralrelaxation in amorphous systems:

$\begin{matrix}{{1 - R_{f}} = {\exp\lbrack {- ( \frac{t}{\tau(T)} )^{\beta}} \rbrack}} & {{Eqn}.6}\end{matrix}$

In Equation 6, β(0<β≤1) is the stretch exponent, τ is the relaxationtime and T is the temperature. As can be seen in FIGS. 14C and 14D,experimental measurements exhibited good agreement with the KWWstretched exponential fits. The values of r calculated by fitting thedata in a double-logarithmic plot (FIGS. 14C and 14D) are presented inFIGS. 14E and 14F, respectively. For the relaxation time, a critical“slowing down” was observed close to the glass transition of SAN aswould be expected. However an important question raised herein is whythe relaxation time of SAN in II and III is so different. It isspeculated that this difference can be attributed to the differentstress constraints on SAN fillers, originated from the elastic matrix.If the filler polymer sizes and grafting are roughly equal, the stressconstraints mainly depend on the content of filler polymers. For lowcontent of filler polymers, an increase stress constrains can beanticipated, resulting in a longer relaxation time.

The approach undertaken in this section attempts to interpret theunderlying mechanism of structural conversion in flexible PU foams 205.It is speculated that the structural converbility of these foams 205 canbe rationalized in terms of three factors: SAN particles, hydrogenbonding between hard- and soft-domains, and adhesion between the cellribs generically resulted from van der Waals interactions Due to thesignificant role of SAN filler polymer's relaxation process in thestructural conversion, the stretched exponential function was employedas a simple way to analyze the structural conversion experiments in theSAN containing flexible PU foams (II and III). The findings aboveprovide valuable insights into the important relationship betweenprocessing temperature and heating time for the optimal design ofauxetic manufacture of flexible PU foams 205.

Auxetic Foams Manufacture

In an attempt to compare the above findings to the practical auxeticmanufacture process, auxetic samples 205 were prepared. It was foundthat sample I cannot be converted to auxetic foam due to its extremepoor structural convertibility from hard-soft domains hydrogen bonding.Thus, the following discussion will only focus on II and III. FIG. 15shows heating time as a function of processing temperature in theauxetic manufacture of foams II and III. The results show that theauxetic manufacture processing condition is clearly related to the SANloading. This observation suggests that simply by increasing the SANloading, the structural convertibility, with regard to the auxeticmanufacturing, can be easily satisfied at least from the view point ofprocess. For given foams 205, the relationship between heating time andprocess temperature is almost consistent with the relaxation time data.This reflects the critical role of SAN relaxation process in auxeticmanufacture.

Moreover, considerable evidence from SEM supports this finding. FIGS.16A through 16D show the SEM images for the auxetic foam 205 with aPoisson's ratio −0.69 (fabricated from III at 150° C. for 30 min). Thetypical re-entrant structure is dearly illustrated in FIG. 16A. Also thestretching SAN particles along the local stress direction are confirmedby the finer SEM images shown in FIGS. 16C and 16D. Thus it is plausiblethat the SAN particles serve as curing agents which could “freeze” there-entrant structure formed when the foam 205 is cooled to below itsglass transition temperature.

It is interesting to note that no matter what role the adhesion plays inthe structural convertibility of flexible PU foams 205, it should beavoided in the fabrication of auxetic foams 205. This is due to theadhesion between the cell ribs causing a significantly retard nonaffinekinematic. FIG. 17 shows a typical SEM image of the sample 205fabricated from III at 150° C. for 60 min (volume compression ratio0.85). Although this sample still presents a similar re-entrantstructure, the finer images show that some of the contact interface iscombined, resulting in a positive Poisson's ratio (ν=0.13).

Based on the results herein, FIG. 18 illustrates a schematic diagram toexplain the process window for the fabrication of auxetic flexible PUfoam 205. During the conversion process, the temperature normally shouldbe higher than the glass transition temperature of SAN filler polymersfor a reasonable process time. Medium compression strain is also neededto force the cell ribs to buckle. It vas also argued that thecombination of high temperature and high compression strain is extremelyharmful for auxetic manufacturing due to the unavoidable adhesionbetween the cell ribs.

As presented herein, several critical questions concerning auxeticmanufacturing of flexible PU foams 205 are answered, based on a relativecomplete understanding of the microstructure for the conventionalflexible PU foams 205 studied herein by the combination of differenttechniques. The significant role of SAN filler polymers in thestructural conversion of the flexible PU foams 205 has been demonstratedfor the first time. This interesting finding indicates that SANcontaining flexible PU foams 205 are excellent choices for use in thefabrication of auxetic PU foams 205. Measurements of % g are acceptableprobes of identification of the softening temperature mentioned in thegeneral instruction for auxetic manufacture of flexible PU foams 205.The stretched exponential function is a simple but useful tool toidentify the optimum processing temperature and heating time in thefabrication process of auxetic PU foams 205.

Fabrication of Auxetic PU Foams

PU foams 205 used for auxetic manufacture may comprise two domains: softdomain with dissolved hard domain and styrene acrylonitrile copolymer(SAN) filler polymers. The soft domain provides PU foams 205 sufficientdeformational ability required for structural conversion, while the harddomain acts as “curing agents” to fix the deformed structure of PU foams205 via the remarkable mobility change of macromolecular chains aroundthe glass transition temperature (T_(g)). This finding can helpelucidate some of the problems reported in the typical“compression-heating-cooling” procedure for production of auxetic PUfoams 205, and also provide an answer to why acetone plays a rolesimilar to an increase in temperature in chemical-mechanical approach.This effect may be due to the strong interactions between SAN and polarsolvents (e.g., acetone, dimethylformamide and chloroform) resulting ina large depression of glass transition temperature of SAN fillerpolymers (T_(g·SAN)).

Carbon dioxide (CO₂) is of growing interest as a solvent in industrialpractice and academic research due to its attractive properties, such asbeing inexpensive, nonflammable, environmentally friendly and easilyremoved from foam 205 products, as well as the tunability ofphysicochemical and mechanical properties (such as density and mobility)by varying pressure and temperature. [59-65] It has been reported thatsubstantial reduction in T_(g) can be expected for polar polymers (e.g.,poly(methyl methacrylate), poly(L-lactide) and acrylonitrile butadienestyrene copolymer) in the presence of dissolved CO₂, [66-69] due toenhanced specific interactions of CO₂ with carbonyl or nitrite groups.[70] Along the same line as Grima et al.'s method. [55] it is,therefore, appropriate to consider CO₂ as a solvent to reduce the glasstransition temperature of SAN and further service to assist thefabrication of auxetic PU foams 205.

FIG. 19 illustrates a schematic diagram of an experimental setup 1900and sample preparation technique. The experimental setup 1900 maycomprise a cylinder 1905 of compressed CO₂ supplying compressed CO₂ gasto a syringe pump 1910. CO₂ from the syringe pump may be used topressure a pressure vessel 1915. Auxetic foam samples 205 were preparedby inserting a raw PU foam specimen 205 into the pressure vessel 1915.The vessel size (V_(h)) was smaller than the initial volume value (V₀)of raw PU foam specimen 205. The volumetric change (VC) of PU foams 205is defined by VC=(V₀−V_(h))/V₀. The vessel 1915 was then filled withcompressed CO₂ at selected temperature (T) and pressure (P). Afterequilibrium was established, the pressure inside the vessel 1915 wasreleased and the foam sample 205 was removed.

FIG. 20A shows the processing window for auxetic manufacture of PU foams205 with compressed CO₂. For these studies, the initial volumetricchange of PU foam specimen 205 was fixed at 0.75 (that is, the foamsample 205 was reduced to 25 percent of its initial volume V₀), and aminimum processing time of 4 h was used to establish thermalequilibrium. In various embodiments, the initial volumetric change maybe less than 0.75. The circles and diamonds in FIG. 20A representunsuccessful and successful processing conditions, respectively. Insetnumbers are measured mean Poisson's ratio in a tensile test with astrain rate of 6 mm/min and maximum strain of 30%. The results clearlyshow that processing temperature can be dramatically reduced by theintroduction of compressed CO₂. Note that under rather moderate pressure(about 5 MPa), auxetic foam can be successfully fabricated even at roomtemperature. FIG. 2013 shows representative plots of Poisson's ratio asa function of engineering strain for raw PU foams 205 and emetic PUfoams 205 (fabricated at 25° C. and 5 MPa with a predefined volumetricchange 0.75). The results show that the Poisson's ratio of raw PU foams205 has a value of +0.38 for small strain and approach +0.5 for largetension strain and 0 for large compression strain. Auxetic foams 205,however, exhibit Poisson's ratio of approximate −0.5 for a wide range ofaxial strains (0 to 0.5), agreeing well with the missing-rib modelproposed by Smith et al. (line in FIG. 20B) [71,72] This finding is verydifferent from previous reports that auxetic PU foams 205 generally showstrain-dependent Poisson's ratio. [19,22,27]

The glass transition temperature of the SAN-CO₂ system was calculated byusing Sanchez-Lacomb equation of state (SL-EoS) [73,74] and applying theGibbs-DiMarzio thermodynamic criterion for glass transition, followingthe thermodynamic framework developed by Condo et al. [75] (the line inFIG. 20A). The processing windows for auxetic manufacture of PU foams205 with compressed CO₂ is well described by the glass-transitiontemperature profile of the SAN-CO₂ system except for some cases atsub-zero temperature and low pressure.

FIG. 20C shows representative scanning electron microscopy (SEM) imagesof raw PU foam 205 and auxetic PU foam 205 (fabricated at 25° C. and 5MPa with a predefined volumetric change 0.75). Compared to the typicalopen-cell structure in raw PU foam 205, auxetic specimen shows agenerally inwardly-buckled structure. [1] Also, a clear shape change inSAN fillers was detected in SEM images taken at high magnification. Itis proposed that when PU foams 205 are exposed to CO₂ under givenprocessing conditions (P and T), CO₂ can dissolve in SAN fillerpolymers, causing a large decrease in glass transition temperature ofSAN (T_(g·SAN-CO2)<T). Thus SAN can undergo a glass transition fromglassy state to rubbery state. In response to external force equilibriumthese fillers can relax quickly to be deformed from their generalspherical shapes, into generally ellipsoidal shapes. After pressurerelease, these fillers can quickly vitrify again (transition from therubbery state to the glassy state) due to the removal of CO₂ and theellipsoidal shape is fixed. A schematic describing the auxeticconversion of PU foams 205 using c6ompressed CO₂ is depicted in FIG.20D.

Next, the effect of processing time on Poisson's ratio was examined.Foam samples 205 with a predefined volumetric change (VC=0.75) werefabricated using CO₂ at 25° C. and 5 MPa for different times. As shownin FIG. 21A, little effect of processing time on Poisson's ratio valuesis found in a wide range of time scale. Manufacturing of auxetic foamcan be completed within seconds. This can be simply explained byconsidering two factors: one is the rapid convection of CO₂ in pores ofPU foams 205, and the other is the extremely short diffusion distance(˜100 μm) for CO₂ gas (FIG. 21A, inset).

Initial volumetric change is another important factor that determinesPoisson's ratio of the resulting samples 205. To study its effect,samples 205 with different VC were fabricated using CO₂ at 25° C. and 5MPa for 10 minutes. As shown in FIG. 21B, applying small volumetricchange (VC≤0.5) would produce samples 205 with almost positive Poisson'sratio values. With the increase in initial volumetric change(0.5<VC≤0.75), Poisson's ratio values decreased approximately linearlyfrom 0 to −0.5. But a further increase in VC (>0.75) led to a slightincrease in Poisson's ratio. This result, in agreement with the previousstudies, suggests a volumetric change of 0.5-0.85.

These results demonstrate that auxetic PU foams 205 can be fabricated atroom temperature by judicious choice of the pressure of CO₂ anddisplayed a unique feature: the Poisson's ratio of auxetic foams 205 isalmost independent of the applied tensile strain up to 50%. Thisfabrication is efficient, economic and environmentally benign, implyingthe potential for large-scale industrial application.

In various embodiments, the pressure vessel 1915 (see FIG. 19 ) maycomprise a mold for a particular shape. As the foam 205 is compressed,the foam 205 conforms to the shape of the mold. Once the tiller polymertransitions from the rubbery state back to the glassy state, the foam205 may retain the shape of the mold after removal from the pressurevessel 1915.

Optimization and Scale Up of Auxetic Foam Manufacturing Process

Approach 1

An existing process, which produced one auxetic foam 205 sheet at atime, used a single fabrication unit in a main heated chamber to producethe auxetic foam sheet. The productivity may be increased by usingmultiple conversion units so that multiple foam 205 sheets may beproduced in one batch operation cycle. Processing time must beoptimized. When the number of fabrication units used is increased fromsingle to multiple units, the temperature and the distribution withinthe main chamber mill change, which will affect the materials relaxationprocess. This in turn still dictate the required time for the auxeticconversion, which is the most crucial process that decides the finalquality of the auxetic foams 205. Temperatures within the multipleconversion units will be carefully measured, and the respectiverelaxation times for the foams 205 in these units will be calculated byusing Kohlrausch, Williams and Watts (KWW) stretched exponentialfunction and modeling fitting [80,81]. The longest relaxation time willbe used for the processing time for multiple sheets conversion process.

Approach 2

Since the substantial solubility, of CO₂ is in the materials used, thepolymer mobility is significantly enhanced and relaxation may take placemore rapidly at much lower temperatures (821, providing severaladvantages over the aforementioned thermal process (Approach 1). Lowerprocessing temperature may result in lower energy consumption andreduced cost. Since PU foams 205 are thermal insulators, heating takes along time during which much of the energy is wasted. Moreover, achievinguniform temperature in large piece or block of PU foam 205 would beextremely difficult. This affects the quality and limits the sue of theauxetic foams 205 that can be fabricated. Using multiple conversionunits with smaller thickness may only partially address the foam 205quality issue and may impose constraints on achievable sheet formfactors for sock fabrication. Studies using lab-scale equipment haveshown that auxetic foams 205 may be produced at near room temperature,which would greatly reduce or even eliminate the energy for heating.Results have suggested that the overall cycle time can be reduced toseveral minutes or even within seconds, instead of hours required forthe thermal conversion process discussed earlier. Thus this technologywould enable manufacturing of auxetic foams 205 of much larger sizes ata fraction of the current costs and time.

FIG. 22 is a schematic illustration of the process according to variousembodiments. Tooling may be designed and machined to hold the regular PUfoams 205 (work piece). The foam 205 work piece may be first compressedbiaxially in the in-plane directions, after which the foam 205 may bepushed down from the out-of-plane direction into the pressure vessel1915 while simultaneously compressed in the third dimension. The vessel1915 may be sealed and carbon dioxide injected. The vessel 1915 may beheld at certain pressure and may be heated if necessary. Auxeticconversion may proceed. Thereafter the CO₂ pressure will be released andthe auxetic foam 205 removed from the vessel 1915.

Application of Auxetic Foams in Prosthetics

Auxetic materials also possess a unique property that would hugelybenefit below-knee (BK) patient. When bent, the foam materials 205 formdoubly curved or domed shapes due to their synclastic curvatureproperties, as shown in FIG. 23A. Socks made of these materials wouldallow for the natural conformity to the limb, especially to the kneeregion throughout flexion/extension, which is crucial so as not toimpede the natural range of motion of the limb. Conventional materialsform saddle-like shapes in the direction perpendicular to knee motion asshown in FIG. 23B, restraining the knee motion and causing discomfort,and may cause crimping of the materials in the lateral direction.

SSMART Sock Manufacturing

Various embodiments may comprise may comprise a prosthetic sockmanufactured from acetic foam 205 as described above. Referred to as theSSMART (smart sock manufactured for amputee rehabilitation and comfort)prosthetic sock, the sock may be directly fabricated from the auxeticfoam 205 sheets. This approach has the advantage of lowered cost andeasy implementation. FIG. 24 illustrates a process for manufacturingsuch socks according to various embodiments. Using a residual limb model(similar to that used for liner fabrication obtained by current standardpractice), the 3D geometry may be captured by a scanner and transferredand convened to a format readable by CAD software such as SolidWorks®. Avirtual “unzip” process may then be performed to unfold the 3D modelthat replicates the external surface of the residual limb into a 2Dpattern. Patterns of this size may be cut out from the auxetic foam 205sheets for sock fabrication. To provide additional structural integrityand reinforcement, two layers of ply cloth, commonly used in some linerfabrication, may be used as backing layers on both sides of the foam205. The assembly may be rolled into the conical shape that accuratelyrepresents the initial limb geometry, followed by sewing and stitchingto complete the sock fabrication. If needed, the seam may bestrengthened by applying a small amount of adhesive.

Net-shape manufacturing of SSMART sock may also be used. Having SSMARTsock fabricated directly into the final desired shape would eliminateseams and improve the structure integrity and durability of the sock.FIG. 25 illustrates an exemplary method according to variousembodiments. The first step may be the synthesis of the regular PU foam205 sock in a closed mold set (mold 1). A metered amount of PU foam 205formulation may be charged into the mold. Ensuing reactive foaming mayproduce the intermediate conventional PU foam 205 sock. Following thisthe mold may be opened and a second male mode (mold 2) may be used toexert the isotropic tri-axial compression, CO₂ may be injected into themold during the auxetic conversion step to lower the processingtemperature and reduce the conversion time, similar to what's discussedin Approach 2 and FIG. 20 above. By changing the geometric parameters ofthe two sets of molds, such as the length, the radius, tapered angle andof the radius of the male mold lid, SSMART socks with designed size,shape and thickness (and/or thickness gradient) may be fabricated.

Summary of Methods for Producing Auxetic Foam

FIG. 26 illustrates a general flow diagram of various embodiments of amethod 2600 for producing an auxetic foam 205. At step 2605, a flexiblefoam 205 may be provided. The foam 205 may have an initial volume andmay comprise a plurality of cells. A bulk matrix polymer of the foam maycomprise a soft domain, and a hard domain. The bulk matrix polymer maycomprise a filler polymer. The foam 205 may be placed into a pressurechamber 1915 at step 2610. The loam 205 may be compressed to acompressed volume that is less than the initial volume. The cells of thefoam 205 may be deformed when the foam 205 is compressed. At step 2615,the compressed foam 205 may be exposed to a compressed gas within thepressure chamber 1915. The pressure chamber 1915 may be maintained at apredetermined temperature and pressure for a predetermined time. Atleast a portion of the compressed gas may dissolve into the filler atstep 2620. The dissolved gas may reduce a glass transition temperatureof the filler polymer such that the filler polymer transitions from aglassy state to a rubbery state. At step 2625, the pressure may berelieved and allowed to reach atmospheric pressure before removing thefoam 205 from the chamber 1915, such that the filler polymer transitionsfrom the rubbery state to the glassy state, thereby fixing the cells ofthe foam 205 in the deformed state.

FIG. 27 illustrates a general flow diagram of various embodiments ofanother method 2700 for producing an auxetic foam 205. At step 2705, aflexible foam 205 may be provided. The foam 205 may have an initialvolume and may comprise a plurality of cells. A bulk matrix polymer ofthe foam may comprise a soft domain, and a hard domain. The bulk matrixpolymer may comprise a filler polymer. The foam 205 may be placed into apressure chamber 1915 at step 2710. The foam 205 may be compressed to acompressed volume that is less than the initial volume. The cells of thefoam 205 may be deformed when the foam 205 is compressed. At step 2715,the compressed foam 205 may be exposed to carbon dioxide within thepressure chamber 1915. The pressure chamber 1915 may be maintained at apredetermined temperature and pressure for a predetermined time. Atleast a portion of the carbon dioxide may dissolve into the filler atstep 2720. The dissolved carbon dioxide may reduce the glass transitiontemperature of the filler polymer such that a shape of the fillerpolymer transitions from generally spherical to generally ellipsoidal.At step 2725, the pressure may be relieved and allowed to reachatmospheric pressure before removing the foam 205 from the chamber 1915,such that the filler polymer retains the generally ellipsoidal shape,thereby fixing the cells of the foam 205 in the deformed state.

FIG. 28 illustrates a general flow diagram of various embodiments of yetanother method 2800 for producing an auxetic foam 205. At step 2805, aflexible foam 205 may be provided. The foam 205 may have an initialvolume and may comprise a plurality of cells. A bulk matrix polymer ofthe foam may comprise a soft domain having a first glass transitiontemperature, and a hard domain having a thermal transition temperaturegreater than the first glass transition temperature. The bulk matrixpolymer may comprise a filler polymer. The filler polymer may have asecond glass transition temperature greater than the first glasstransition temperature and less than the thermal transition temperature.The second glass transition temperature may be reduced by dissolving acompressed gas in the filler. The foam 205 may be placed into a pressurechamber 1915 at step 2810. The foam 205 may be compressed to acompressed volume that is less than the initial volume. The cells of thefoam 205 may be deformed when the foam 205 is compressed. The compressedfoam 205 may be exposed to a compressed gas within the pressure chamber1915. At step 2815, the compressed foam 205 may be heated within thepressure chamber 1915 to a temperature greater than the second glasstransition temperature and less than the thermal transition temperature.The compressed foam 205 may be held at the temperature for apredetermined time. The compressed foam 205 may be allowed to cool atstep 2820 while remaining in the pressure chamber 1915 to a temperatureless than the second glass transition temperature. At step 2825, thefoam 205 may be removed from the pressure chamber 1915, thereby fixingthe cells of the foam 205 in the deformed state.

EXAMPLES Example 1

One commercial open-cell, flexible PU foam 205 (SAN content: 31.6 wt %)[58] with nominal cell diameter of 480 μm was employed in this study.The density of the foam was 48.1 kg/m³. The foam 205 was machined intocylindrical specimens with predefined dimensions (see Table 1) using aCO₂ laser (VersaLASER, Universal Laser Systems). One of these machinedfoams 205 was then inserted into a pressure reactor 1915 (CL-1, HighPressure Equipment Company), as shown in FIG. 19 . The vessel 1915 waspurged with CO₂ and then CO₂ was fed by a high-pressure ISCO syringepump 1910 (500HP, Teledyne Technologies, Inc.) that also was used tomaintain the constant system pressure. After equilibrium wasestablished, the pressure was released. Finally, the foams 205 weretaken out of the reactor 1915. Poisson's ratio was determined asdescribed previously.

TABLE 1 Radial and axial compression for various volumetric changes.Volumetric Initial size (cm) Imposed size (cm) change Diameter HeightDiameter Height 0.4 8.9 10 7.5 8.4 0.45 9.1 10 7.5 8.2 0.5 9.4 10 7.57.9 0.55 9.8 10 7.5 7.7 0.6 10.2 10 7.5 7.4 0.65 10.6 10 7.5 7 0.7 11.210 7.5 6.7 0.75 11.9 10 7.5 6.3 0.8 12.8 10 7.5 5.8 0.85 14.1 10 7.5 5.3

Calculation of Poisson's Ratio Using Missing-rib Model [72] (see insertin FIG. 20B)

Engineering strain may be calculated using Equation 7:

$\begin{matrix}{\varepsilon_{x} = {4\lbrack {\frac{\cos( {\zeta_{0} - \phi_{0} + {{\Delta\phi}( {\kappa - 1} )}} )}{\cos( {\zeta_{0} - \phi_{0}} )} - 1} \rbrack}} & {{Eqn}.7}\end{matrix}$

κ=Δξ/Δϕ is a measure of relative deformation between the ξ and ϕsprings. Here, ξ 90°, ϕ=45° and κ=0.53.

Poisson's ratio may be calculated using Equation 8:

$\begin{matrix}{\nu_{xy} = \frac{\{ {{\cos\lbrack {\zeta_{0} - \phi_{0} + {\Delta{\phi( {\kappa - 1} )}}} \rbrack} - {\cos( {\zeta_{0} - \phi_{0}} )}} \}\sin\phi_{0}}{( {{\sin\phi} - {\sin\phi_{0}}} ){\cos( {\zeta_{0} - \phi_{0}} )}}} & {{Eqn}.8}\end{matrix}$

Calculation of the Glass Transition Temperature of the SAN-CO₂ System

The Sanchez-Lacombe equation of state (SL EoS) [73,74] is shown inEquation 9:

$\begin{matrix}{{{\overset{\sim}{\rho}}^{2} + \overset{˜}{P} + {\overset{\sim}{T}\lbrack {{\ln( {1 - \overset{\sim}{\rho}} )} + {( {1 - \frac{1}{r}} )\overset{\sim}{\rho}}} \rbrack}} = 0} & {{Eqn}.9}\end{matrix}$

where {tilde over (T)}, {tilde over (P)} and {tilde over (ρ)} aredefined as:

${\overset{˜}{T} = \frac{T}{T^{\star}}};{\overset{\sim}{P} = \frac{P}{P^{\star}}};{\overset{\sim}{\rho} = \frac{\rho}{\rho^{\star}}}$

and T*, P* and ρ* are the scaling parameters. Table 2 shows the scalingparameters of CO₂ and styrene acrylonitrile copolymer (SAN) with anacrylonitrile (AN) content of 30 wt %.

TABLE 2 Scaling parameters of CO₂ and styrene acrylonitrile copolymer(SAN) with an acrylonitrile (AN) content of 30 wt %. P* (MPa) ρ* (kg ·m⁻³) T* (K) Ref. CO₂ 611.00 1413.3 278.5 4 SAN 588.68 1174.2 731.3 5

The number of lattice sites occupied by a molecule, r, is given byEquation 10:

$\begin{matrix}{r = \frac{M_{W}P^{\star}}{{RT}\rho^{\star}}} & {{Eqn}.10}\end{matrix}$

where R is the gas constant and Mw is molecular weight. For mixtures,the mixing rules presented in Equations 11 and 12 are used:

$\begin{matrix}{{P^{\star} = {\sum\limits_{i}{\sum\limits_{j}{\phi_{i}\phi_{j}P_{ij}^{\star}}}}};{T^{\star} = {P \star {\sum\limits_{i}\frac{\phi_{i}^{\circ}T_{i}^{\star}}{P_{i}^{\star}}}}};{\frac{1}{r} = {\sum\limits_{i}\frac{\phi_{i}^{\circ}}{r_{i}}}}} & {{Eqn}.11}\end{matrix}$ $\begin{matrix}{{{{where}P_{ij}^{\star}} = {( {1 - \kappa_{ij}} )\sqrt{P_{i}^{\star}P_{j}^{\star}}}};} & {{Eqn}.12}\end{matrix}$${\phi_{i} = {( \frac{wi}{\rho_{i}^{\star}} )/{\sum\limits_{j}\frac{w_{j}}{\rho_{j}^{\star}}}}};$$\phi_{i}^{\circ} = {\frac{\phi_{i}T_{i}^{\star}}{P_{i}^{\star}}/{\sum\limits_{j}\frac{\phi_{j}T_{j}^{\star}}{P_{j}^{\star}}}}$

where ϕ₁ is the volume fraction of component f and k_(ij) is the binaryinteraction parameter. The system entropy (S) of a binary system can bederived from SL EoS [75,78] as shown in Equation 13:

$\begin{matrix}{\frac{S}{rNK} = {{( {1 - \frac{1}{\overset{\sim}{\rho}}} ){\ln( {1 - \overset{\sim}{\rho}} )}} - \frac{\ln\overset{\sim}{\rho}}{r} - {( \frac{\phi_{1}}{r_{1}} ){\ln( \frac{\phi_{1}}{r_{1}} )}} - {( \frac{\phi_{2}}{r_{2}} ){\ln( \frac{\phi_{2}}{r_{2}} )}} - \frac{\ln( {2/z} )}{r} + \frac{1}{r} - 1 + {( \frac{\phi_{1}}{r_{1}} ){( {r_{1} - 2} )\lbrack {\frac{f_{1}{\Delta\varepsilon}_{1}}{kT} - {\ln( {1 - f_{1}} )}} \rbrack}} + {( \frac{\phi_{2}}{r_{2}} ){( {r_{2} - 2} )\lbrack {\frac{f_{2}{\Delta\varepsilon}_{2}}{kT} - {\ln( {1 - f_{2}} )}} \rbrack}}}} & {{Eqn}.13}\end{matrix}$

where z is the lattice coordination number and Δε_(i) represents theincrease of intramolecular energy. f₁ is the equilibrium fraction offlexed bonds given by Equation 14:

$\begin{matrix}{f_{i} = \frac{( {z - 2} ){\exp( {{- {\Delta\varepsilon}_{i}}/{kT}} )}}{1 + {( {z - 2} ){\exp( {{\Delta\varepsilon}_{i}/{kT}} )}}}} & {{Eqn}.14}\end{matrix}$

The Δε₁ of CO₂ was assumed to be zero. [78] z=4 and Δε₂/k=598.4 K whichwere obtained by the fitting curve of from glass transition behavior forpure SAN (FIG. 29 ). [79] The interaction parameter k_(ij)=−0.029 wasobtained by fitting to experimental solubility of CO₂ in SAN at 25° C.and 6.55 MPa (FIG. 30 ). [70] Thus, T_(g) of the SAN-CO₂ system can becalculated by setting the system entropy equal to zero (Equation 13),according to the Gibbs-Di Marzio criterion (see FIG. 20(a)).

Example 2

Quasi-Static Compression

Compression tests on foams 205 were carried out using a Shimadzu DV-201fitted with a 1000 N load cell at 23° C. The specimens 205 werecylindrical with diameter and height. The compression rate in all testswas 6 mm/min. The energy absorbed per unit volume at a certain peakstress, was obtained by calculating the area under the stress-straincurve up the peak stress.

Cyclic Loading Experiments

In this test, each specimen 205 was cyclically compressed in the loadframe to a set strain endpoint for 100 cycles. The compression rate inall tests was 6 mm/min.

Dynamic Mechanical Analyzer (DMA) Test

Dynamic properties of foams 205 were examined using a dynamic mechanicalanalyzer (DMA, TA Instruments DMA Q800) equipped with a parallel-platecompression clamp with a diameter of 15 mm. Test were performed incompression mode at 23° C. at a side frequency range from 0.01 to 100 HZwith a dynamic strain of 2% with an appropriate static preload of 0.05N.

Results

The test results for Example 2 are present in FIGS. 31 through 36 . FIG.31 shows the effect of volumetric change on the Poisson's ratio ofauxetic PU foams 205 during auxetic manufacture with compressed CO₂.FIG. 32 shows the stress-strain plots from the compressions tests offoams 205 with different volumetric change ratios. FIG. 33 shows theenergy adsorption (W) versus peak stress curve of foams 205 withdifferent volumetric change ratios, where W is calculated using Equation15:

W=∫ ₀ ^(z)σ(ε)de  Eqn. 15

FIG. 34 shows the cushioning coefficient (C) versus stress curve offoams 205 with different volumetric change ratios, where C is calculatedusing Equation 16:

$\begin{matrix}{C = \frac{\sigma}{\int_{0}^{\epsilon}{{\sigma(e)}{de}}}} & {{Eqn}.16}\end{matrix}$

FIG. 35 shows the energy dissipation of foams 205 with differentvolumetric change ratios (and Poisson's ratios). FIG. 36 shows dynamicproperties of raw PU foam 205 and auxetic PU foam 205 Higher tan deltaindicates higher energy dissipation capability.

Definition of Claim Terms

Auxetic: Materials having a Poisson's ratio less than zero. Auxeticmaterials expand laterally during stretching and shrink undercompression.

Cell: In a flexible foam material, void spaces within the otherwisesolid foam material.

Deformed: For a foam cell, the point at which the cell walls (or struts)buckle inward into the cell when pressure is applied to the foam.

Foam: A substance formed by trapping pockets of gas within a liquid orsolid. The pockets of gas form cells within the foam.

Glass transition temperature: The temperature below which a polymerbecomes more hard and brittle and above which the polymer is more softand flexible.

Glassy state: A hard, brittle state of a polymer material that is belowits glass transition temperature.

Hard domain: A polymer material below its glass transition temperature.

Poisson's ratio: a physical property of a material calculated as thefraction of expansion divided by the fraction of compression for amaterial that is compressed in one direction while expanding in twoother directions.

Rubbery state: a soft, flexible state of a polymer material that isabove its glass transition temperature.

Soft domain: A polymer material above its glass transition temperature.

Thermal transition temperature: A temperature at which a physicalproperty of a material changes, including but not limited to a glasstransition temperature, a melting temperature, or a solidificationtemperature.

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1. (canceled)
 2. The material system of claim 21, wherein the fillerpolymer comprises styrene acrylonitrile copolymer. 3-8. (canceled) 9.The material system of claim 29, wherein the compressed gas is primarilycarbon dioxide or nitrogen.
 10. (canceled)
 11. (canceled)
 12. Thematerial system of claim 21, wherein varying a concentration of thefiller polymer in the foam causes variation of mechanical properties ofthe foam. 13-20. (canceled)
 21. A material system for the production ofauxetic foam, comprising: a bulk matrix, comprising: a soft domaincomprising a polymer chain and having a first glass transitiontemperature; a hard domain covalently or non-covalently linked to thepolymer chain of the soft domain and having a thermal transitiontemperature greater than the first glass transition temperature; and afiller polymer dispersed in the bulk matrix and having a second glasstransition temperature greater than the first glass transitiontemperature and less than the thermal transition temperature.
 22. Thematerial system of claim 21, wherein the bulk matrix and filler polymertogether comprise a multiphase, multicomponent polymer foam.
 23. Thematerial system of claim 22, wherein the polymer foam comprises apolyurethane foam.
 24. The material system of claim 21, wherein thepolymer chain comprises polymeric polyols.
 25. The material system ofclaim 21, wherein the filler polymer comprises styrene acrylonitrile,polyether sulfone, polysulfone, cyclic olefin copolymers,acrylonitrile-butadiene-styrene, poly(p-phenylene oxide), poly(etherketone), poly(ether ketone), poly(ether ketone ketone), or combinationsthereof.
 26. The material system of claim 25, wherein the filler polymeris chosen based on the solubility of carbon dioxide or nitrogen in thefiller polymer.
 27. The material system of claim 21, wherein the bulkmatrix is elastic.
 28. The material system of claim 21, wherein the bulkmatrix deforms when subjected to a mechanical compressive force.
 29. Thematerial system of claim 21, wherein the filler polymer has a shape thatchanges from generally spherical to generally ellipsoidal when the bulkmatrix is subjected to a heating step and/or when the bulk matrix isexposed to a compressed gas.